Page 218 - Steam Turbines Design, Applications, and Rerating

Campbell, Goodman, and SAFE Diagrams for Steam Turbine Blades 197

the phase relationship between the exciting force and blade motion,
and the damping of the blade-shroud structure. If there were no
damping in the blades, the vibration amplitude would continuously
build up until failure occurred. This sometimes happens, but the usual
case is that the damping is sufficient to prevent dangerous levels of
vibration to build up.
If the Campbell diagram shows that a resonance exists in the oper-
ating speed range, several design changes can be made to avoid it. The
blade frequency can be changed by going to a stronger, stiffer blade sec-
tion. If the resonance is with nozzle-passing frequency, the number of
nozzles can be changed to avoid it. If it is impossible to avoid a reso-
nance, then the steady stresses must be reduced to a level at which it
is acceptable to operate during resonance. Methods for reducing steady
stresses were discussed in the aforementioned Goodman diagram
section.
A few words of caution are in order when using the Campbell dia-
gram to evaluate blading. It should be noted that it is still difficult to
accurately determine blade natural frequencies, even with improved
analytical and test methods. This is especially true with short blading.
Manufacturing tolerance and assembly variations can account for as
much as a ±10% variation in blade natural frequency. Large avoidance
margins must therefore be utilized to ensure that no interference
exists.
In addition, all interferences shown on the Campbell diagram do not
mean that the blade vibratory stresses will be high and that blade fail-
ure is imminent. Not all interferences are true resonances. The Camp-
bell diagram only compares if the exciting force frequency and blade
natural frequency match, but it does not compare if the shape of the
exciting force and blade natural frequency match. For true resonance
to exist, both the frequency and shape must match. Also, even if true
resonance does occur, this does not mean that blade failure is certain.
The level of vibratory stress occurring at resonance depends on the
strength of the exciting force, the amount of damping in the structure,
and the overall stress levels in the blading.
Figure 11.3 illustrates the near-impossibility of avoiding all reso-
nances in the operating speed range.

11.4 SAFE Diagram—Evaluation Tool for
Packeted Bladed Disk Assembly
The Campbell diagram is a two-dimensional projection of a three-
dimensional surface; hence, it does not retain all the information for
evaluation. Another two-dimensional projection of the same surface
contains more information for easier evaluation. This diagram, refined